LMFDB - Embedded newform 195.6.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,6,Mod(1,195)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("195.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 195 = 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	195.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.2748448635$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 67x^{2} + 57x + 250 $ x^4 - x^3 - 67x^2 + 57x + 250
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$7.99762$ of defining polynomial
Character	$\chi$	$=$	195.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.99762 q^{2} -9.00000 q^{3} +16.9667 q^{4} +25.0000 q^{5} -62.9786 q^{6} +66.9584 q^{7} -105.197 q^{8} +81.0000 q^{9} +174.940 q^{10} -721.813 q^{11} -152.700 q^{12} +169.000 q^{13} +468.549 q^{14} -225.000 q^{15} -1279.07 q^{16} +781.362 q^{17} +566.807 q^{18} -2302.21 q^{19} +424.167 q^{20} -602.625 q^{21} -5050.97 q^{22} +3338.48 q^{23} +946.777 q^{24} +625.000 q^{25} +1182.60 q^{26} -729.000 q^{27} +1136.06 q^{28} -8038.05 q^{29} -1574.46 q^{30} -4569.45 q^{31} -5584.10 q^{32} +6496.31 q^{33} +5467.67 q^{34} +1673.96 q^{35} +1374.30 q^{36} -12449.8 q^{37} -16110.0 q^{38} -1521.00 q^{39} -2629.94 q^{40} -10075.1 q^{41} -4216.94 q^{42} +11914.5 q^{43} -12246.8 q^{44} +2025.00 q^{45} +23361.4 q^{46} -10204.7 q^{47} +11511.6 q^{48} -12323.6 q^{49} +4373.51 q^{50} -7032.26 q^{51} +2867.37 q^{52} -13656.0 q^{53} -5101.26 q^{54} -18045.3 q^{55} -7043.85 q^{56} +20719.8 q^{57} -56247.2 q^{58} +7164.16 q^{59} -3817.50 q^{60} -40796.2 q^{61} -31975.2 q^{62} +5423.63 q^{63} +1854.71 q^{64} +4225.00 q^{65} +45458.7 q^{66} +29213.1 q^{67} +13257.1 q^{68} -30046.3 q^{69} +11713.7 q^{70} +71183.1 q^{71} -8520.99 q^{72} +37309.7 q^{73} -87119.2 q^{74} -5625.00 q^{75} -39060.8 q^{76} -48331.4 q^{77} -10643.4 q^{78} +102581. q^{79} -31976.6 q^{80} +6561.00 q^{81} -70501.6 q^{82} +91063.1 q^{83} -10224.6 q^{84} +19534.0 q^{85} +83373.1 q^{86} +72342.4 q^{87} +75932.8 q^{88} +20540.8 q^{89} +14170.2 q^{90} +11316.0 q^{91} +56642.9 q^{92} +41125.0 q^{93} -71408.5 q^{94} -57555.1 q^{95} +50256.9 q^{96} +38094.6 q^{97} -86235.7 q^{98} -58466.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} - 36 q^{3} + 9 q^{4} + 100 q^{5} + 27 q^{6} - 87 q^{7} - 183 q^{8} + 324 q^{9} - 75 q^{10} - 631 q^{11} - 81 q^{12} + 676 q^{13} + 1344 q^{14} - 900 q^{15} - 351 q^{16} - 599 q^{17} - 243 q^{18}+ \cdots - 51111 q^{99}+O(q^{100}) $ 4 * q - 3 * q^2 - 36 * q^3 + 9 * q^4 + 100 * q^5 + 27 * q^6 - 87 * q^7 - 183 * q^8 + 324 * q^9 - 75 * q^10 - 631 * q^11 - 81 * q^12 + 676 * q^13 + 1344 * q^14 - 900 * q^15 - 351 * q^16 - 599 * q^17 - 243 * q^18 - 4 * q^19 + 225 * q^20 + 783 * q^21 - 1920 * q^22 + 4459 * q^23 + 1647 * q^24 + 2500 * q^25 - 507 * q^26 - 2916 * q^27 - 1048 * q^28 - 5274 * q^29 + 675 * q^30 + 2642 * q^31 - 199 * q^32 + 5679 * q^33 + 15794 * q^34 - 2175 * q^35 + 729 * q^36 - 30189 * q^37 - 26408 * q^38 - 6084 * q^39 - 4575 * q^40 - 8947 * q^41 - 12096 * q^42 - 15928 * q^43 - 29676 * q^44 + 8100 * q^45 - 14272 * q^46 - 28242 * q^47 + 3159 * q^48 - 52335 * q^49 - 1875 * q^50 + 5391 * q^51 + 1521 * q^52 - 38021 * q^53 + 2187 * q^54 - 15775 * q^55 - 2480 * q^56 + 36 * q^57 - 115014 * q^58 - 17200 * q^59 - 2025 * q^60 - 67669 * q^61 - 99568 * q^62 - 7047 * q^63 - 69455 * q^64 + 16900 * q^65 + 17280 * q^66 + 8832 * q^67 - 45054 * q^68 - 40131 * q^69 + 33600 * q^70 + 128633 * q^71 - 14823 * q^72 - 48706 * q^73 + 35026 * q^74 - 22500 * q^75 - 66036 * q^76 - 25533 * q^77 + 4563 * q^78 + 72799 * q^79 - 8775 * q^80 + 26244 * q^81 - 191810 * q^82 + 15868 * q^83 + 9432 * q^84 - 14975 * q^85 + 247924 * q^86 + 47466 * q^87 + 5204 * q^88 - 45713 * q^89 - 6075 * q^90 - 14703 * q^91 + 271408 * q^92 - 23778 * q^93 - 51544 * q^94 - 100 * q^95 + 1791 * q^96 - 148949 * q^97 + 10289 * q^98 - 51111 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	6.99762	1.23702	0.618508	−	0.785778i	$-0.287737\pi$
$2$	6.99762	1.23702	0.618508	+	0.785778i	$0.287737\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.9667	0.530209
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−62.9786	−0.714192
$7$	66.9584	0.516487	0.258244	−	0.966080i	$-0.416856\pi$
$7$	66.9584	0.516487	0.258244	+	0.966080i	$0.416856\pi$
$8$	−105.197	−0.581139
$9$	81.0000	0.333333
$10$	174.940	0.553210
$11$	−721.813	−1.79863	−0.899317	−	0.437298i	$-0.855936\pi$
$11$	−721.813	−1.79863	−0.899317	+	0.437298i	$0.855936\pi$
$12$	−152.700	−0.306116
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	468.549	0.638903
$15$	−225.000	−0.258199
$16$	−1279.07	−1.24909
$17$	781.362	0.655737	0.327869	−	0.944723i	$-0.393670\pi$
$17$	781.362	0.655737	0.327869	+	0.944723i	$0.393670\pi$
$18$	566.807	0.412339
$19$	−2302.21	−1.46305	−0.731527	−	0.681813i	$-0.761192\pi$
$19$	−2302.21	−1.46305	−0.731527	+	0.681813i	$0.761192\pi$
$20$	424.167	0.237117
$21$	−602.625	−0.298194
$22$	−5050.97	−2.22494
$23$	3338.48	1.31592	0.657959	−	0.753054i	$-0.271420\pi$
$23$	3338.48	1.31592	0.657959	+	0.753054i	$0.271420\pi$
$24$	946.777	0.335521
$25$	625.000	0.200000
$26$	1182.60	0.343087
$27$	−729.000	−0.192450
$28$	1136.06	0.273846
$29$	−8038.05	−1.77482	−0.887412	−	0.460976i	$-0.847499\pi$
$29$	−8038.05	−1.77482	−0.887412	+	0.460976i	$0.847499\pi$
$30$	−1574.46	−0.319396
$31$	−4569.45	−0.854003	−0.427001	−	0.904251i	$-0.640430\pi$
$31$	−4569.45	−0.854003	−0.427001	+	0.904251i	$0.640430\pi$
$32$	−5584.10	−0.964002
$33$	6496.31	1.03844
$34$	5467.67	0.811158
$35$	1673.96	0.230980
$36$	1374.30	0.176736
$37$	−12449.8	−1.49506	−0.747531	−	0.664227i	$-0.768761\pi$
$37$	−12449.8	−1.49506	−0.747531	+	0.664227i	$0.768761\pi$
$38$	−16110.0	−1.80982
$39$	−1521.00	−0.160128
$40$	−2629.94	−0.259893
$41$	−10075.1	−0.936029	−0.468014	−	0.883721i	$-0.655030\pi$
$41$	−10075.1	−0.936029	−0.468014	+	0.883721i	$0.655030\pi$
$42$	−4216.94	−0.368871
$43$	11914.5	0.982663	0.491332	−	0.870973i	$-0.336510\pi$
$43$	11914.5	0.982663	0.491332	+	0.870973i	$0.336510\pi$
$44$	−12246.8	−0.953652
$45$	2025.00	0.149071
$46$	23361.4	1.62781
$47$	−10204.7	−0.673837	−0.336918	−	0.941534i	$-0.609385\pi$
$47$	−10204.7	−0.673837	−0.336918	+	0.941534i	$0.609385\pi$
$48$	11511.6	0.721161
$49$	−12323.6	−0.733241
$50$	4373.51	0.247403
$51$	−7032.26	−0.378590
$52$	2867.37	0.147053
$53$	−13656.0	−0.667783	−0.333891	−	0.942612i	$-0.608362\pi$
$53$	−13656.0	−0.667783	−0.333891	+	0.942612i	$0.608362\pi$
$54$	−5101.26	−0.238064
$55$	−18045.3	−0.804373
$56$	−7043.85	−0.300151
$57$	20719.8	0.844694
$58$	−56247.2	−2.19549
$59$	7164.16	0.267939	0.133969	−	0.990985i	$-0.457228\pi$
$59$	7164.16	0.267939	0.133969	+	0.990985i	$0.457228\pi$
$60$	−3817.50	−0.136899
$61$	−40796.2	−1.40377	−0.701884	−	0.712291i	$-0.747658\pi$
$61$	−40796.2	−1.40377	−0.701884	+	0.712291i	$0.747658\pi$
$62$	−31975.2	−1.05642
$63$	5423.63	0.172162
$64$	1854.71	0.0566013
$65$	4225.00	0.124035
$66$	45458.7	1.28457
$67$	29213.1	0.795042	0.397521	−	0.917593i	$-0.369871\pi$
$67$	29213.1	0.795042	0.397521	+	0.917593i	$0.369871\pi$
$68$	13257.1	0.347678
$69$	−30046.3	−0.759745
$70$	11713.7	0.285726
$71$	71183.1	1.67583	0.837916	−	0.545798i	$-0.183774\pi$
$71$	71183.1	1.67583	0.837916	+	0.545798i	$0.183774\pi$
$72$	−8520.99	−0.193713
$73$	37309.7	0.819435	0.409717	−	0.912213i	$-0.365627\pi$
$73$	37309.7	0.819435	0.409717	+	0.912213i	$0.365627\pi$
$74$	−87119.2	−1.84942
$75$	−5625.00	−0.115470
$76$	−39060.8	−0.775724
$77$	−48331.4	−0.928971
$78$	−10643.4	−0.198081
$79$	102581.	1.84927	0.924634	−	0.380857i	$-0.124371\pi$
$79$	102581.	1.84927	0.924634	+	0.380857i	$0.124371\pi$
$80$	−31976.6	−0.558609
$81$	6561.00	0.111111
$82$	−70501.6	−1.15788
$83$	91063.1	1.45093	0.725466	−	0.688258i	$-0.241624\pi$
$83$	91063.1	1.45093	0.725466	+	0.688258i	$0.241624\pi$
$84$	−10224.6	−0.158105
$85$	19534.0	0.293255
$86$	83373.1	1.21557
$87$	72342.4	1.02470
$88$	75932.8	1.04526
$89$	20540.8	0.274880	0.137440	−	0.990510i	$-0.456113\pi$
$89$	20540.8	0.274880	0.137440	+	0.990510i	$0.456113\pi$
$90$	14170.2	0.184403
$91$	11316.0	0.143248
$92$	56642.9	0.697711
$93$	41125.0	0.493059
$94$	−71408.5	−0.833547
$95$	−57555.1	−0.654297
$96$	50256.9	0.556567
$97$	38094.6	0.411088	0.205544	−	0.978648i	$-0.434104\pi$
$97$	38094.6	0.411088	0.205544	+	0.978648i	$0.434104\pi$
$98$	−86235.7	−0.907031
$99$	−58466.8	−0.599545
$100$	10604.2	0.106042
$101$	−50607.7	−0.493643	−0.246822	−	0.969061i	$-0.579386\pi$
$101$	−50607.7	−0.493643	−0.246822	+	0.969061i	$0.579386\pi$
$102$	−49209.1	−0.468322
$103$	−62651.4	−0.581885	−0.290943	−	0.956740i	$-0.593969\pi$
$103$	−62651.4	−0.581885	−0.290943	+	0.956740i	$0.593969\pi$
$104$	−17778.4	−0.161179
$105$	−15065.6	−0.133356
$106$	−95559.8	−0.826058
$107$	53587.6	0.452486	0.226243	−	0.974071i	$-0.427356\pi$
$107$	53587.6	0.452486	0.226243	+	0.974071i	$0.427356\pi$
$108$	−12368.7	−0.102039
$109$	−20492.4	−0.165206	−0.0826031	−	0.996583i	$-0.526323\pi$
$109$	−20492.4	−0.165206	−0.0826031	+	0.996583i	$0.526323\pi$
$110$	−126274.	−0.995023
$111$	112049.	0.863175
$112$	−85644.1	−0.645138
$113$	−105210.	−0.775109	−0.387554	−	0.921847i	$-0.626680\pi$
$113$	−105210.	−0.775109	−0.387554	+	0.921847i	$0.626680\pi$
$114$	144990.	1.04490
$115$	83461.9	0.588496
$116$	−136379.	−0.941028
$117$	13689.0	0.0924500
$118$	50132.1	0.331444
$119$	52318.7	0.338680
$120$	23669.4	0.150049
$121$	359962.	2.23508
$122$	−285477.	−1.73648
$123$	90675.8	0.540416
$124$	−77528.3	−0.452800
$125$	15625.0	0.0894427
$126$	37952.5	0.212968
$127$	−294738.	−1.62153	−0.810767	−	0.585369i	$-0.800950\pi$
$127$	−294738.	−1.62153	−0.810767	+	0.585369i	$0.800950\pi$
$128$	191670.	1.03402
$129$	−107231.	−0.567341
$130$	29564.9	0.153433
$131$	−63843.9	−0.325043	−0.162522	−	0.986705i	$-0.551963\pi$
$131$	−63843.9	−0.325043	−0.162522	+	0.986705i	$0.551963\pi$
$132$	110221.	0.550591
$133$	−154152.	−0.755648
$134$	204422.	0.983480
$135$	−18225.0	−0.0860663
$136$	−82197.3	−0.381075
$137$	170391.	0.775614	0.387807	−	0.921741i	$-0.373233\pi$
$137$	170391.	0.775614	0.387807	+	0.921741i	$0.373233\pi$
$138$	−210252.	−0.939817
$139$	−372319.	−1.63448	−0.817238	−	0.576300i	$-0.804496\pi$
$139$	−372319.	−1.63448	−0.817238	+	0.576300i	$0.804496\pi$
$140$	28401.5	0.122468
$141$	91842.1	0.389040
$142$	498112.	2.07303
$143$	−121986.	−0.498851
$144$	−103604.	−0.416362
$145$	−200951.	−0.793726
$146$	261079.	1.01365
$147$	110912.	0.423337
$148$	−211232.	−0.792695
$149$	−519080.	−1.91544	−0.957720	−	0.287702i	$-0.907109\pi$
$149$	−519080.	−1.91544	−0.957720	+	0.287702i	$0.907109\pi$
$150$	−39361.6	−0.142838
$151$	111177.	0.396800	0.198400	−	0.980121i	$-0.436426\pi$
$151$	111177.	0.396800	0.198400	+	0.980121i	$0.436426\pi$
$152$	242186.	0.850237
$153$	63290.3	0.218579
$154$	−338205.	−1.14915
$155$	−114236.	−0.381922
$156$	−25806.3	−0.0849014
$157$	−259512.	−0.840249	−0.420125	−	0.907466i	$-0.638014\pi$
$157$	−259512.	−0.840249	−0.420125	+	0.907466i	$0.638014\pi$
$158$	717824.	2.28757
$159$	122904.	0.385544
$160$	−139602.	−0.431115
$161$	223539.	0.679655
$162$	45911.4	0.137446
$163$	224126.	0.660730	0.330365	−	0.943853i	$-0.392828\pi$
$163$	224126.	0.660730	0.330365	+	0.943853i	$0.392828\pi$
$164$	−170941.	−0.496291
$165$	162408.	0.464405
$166$	637225.	1.79483
$167$	604099.	1.67617	0.838083	−	0.545542i	$-0.183676\pi$
$167$	604099.	1.67617	0.838083	+	0.545542i	$0.183676\pi$
$168$	63394.6	0.173292
$169$	28561.0	0.0769231
$170$	136692.	0.362761
$171$	−186479.	−0.487684
$172$	202150.	0.521017
$173$	174723.	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$173$	174723.	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$174$	506225.	1.26756
$175$	41849.0	0.103297
$176$	923246.	2.24665
$177$	−64477.4	−0.154694
$178$	143737.	0.340031
$179$	−614608.	−1.43372	−0.716862	−	0.697215i	$-0.754422\pi$
$179$	−614608.	−1.43372	−0.716862	+	0.697215i	$0.754422\pi$
$180$	34357.5	0.0790389
$181$	569981.	1.29319	0.646597	−	0.762832i	$-0.276191\pi$
$181$	569981.	1.29319	0.646597	+	0.762832i	$0.276191\pi$
$182$	79184.8	0.177200
$183$	367166.	0.810466
$184$	−351199.	−0.764731
$185$	−311246.	−0.668612
$186$	287777.	0.609922
$187$	−563997.	−1.17943
$188$	−173140.	−0.357274
$189$	−48812.6	−0.0993980
$190$	−402749.	−0.809376
$191$	254138.	0.504064	0.252032	−	0.967719i	$-0.418901\pi$
$191$	254138.	0.504064	0.252032	+	0.967719i	$0.418901\pi$
$192$	−16692.4	−0.0326788
$193$	27615.3	0.0533650	0.0266825	−	0.999644i	$-0.491506\pi$
$193$	27615.3	0.0533650	0.0266825	+	0.999644i	$0.491506\pi$
$194$	266572.	0.508522
$195$	−38025.0	−0.0716115
$196$	−209090.	−0.388771
$197$	883788.	1.62249	0.811246	−	0.584705i	$-0.198790\pi$
$197$	883788.	1.62249	0.811246	+	0.584705i	$0.198790\pi$
$198$	−409129.	−0.741646
$199$	374638.	0.670624	0.335312	−	0.942107i	$-0.391158\pi$
$199$	374638.	0.670624	0.335312	+	0.942107i	$0.391158\pi$
$200$	−65748.4	−0.116228
$201$	−262918.	−0.459018
$202$	−354134.	−0.610645
$203$	−538214.	−0.916674
$204$	−119314.	−0.200732
$205$	−251877.	−0.418605
$206$	−438410.	−0.719802
$207$	270417.	0.438639
$208$	−216162.	−0.346435
$209$	1.66176e6	2.63150
$210$	−105424.	−0.164964
$211$	−1.14233e6	−1.76638	−0.883191	−	0.469014i	$-0.844609\pi$
$211$	−1.14233e6	−1.76638	−0.883191	+	0.469014i	$0.844609\pi$
$212$	−231698.	−0.354064
$213$	−640647.	−0.967543
$214$	374986.	0.559732
$215$	297863.	0.439460
$216$	76688.9	0.111840
$217$	−305963.	−0.441082
$218$	−143398.	−0.204363
$219$	−335787.	−0.473101
$220$	−306169.	−0.426486
$221$	132050.	0.181869
$222$	784073.	1.06776
$223$	−29633.8	−0.0399049	−0.0199524	−	0.999801i	$-0.506351\pi$
$223$	−29633.8	−0.0399049	−0.0199524	+	0.999801i	$0.506351\pi$
$224$	−373902.	−0.497895
$225$	50625.0	0.0666667
$226$	−736223.	−0.958822
$227$	143145.	0.184379	0.0921897	−	0.995741i	$-0.470613\pi$
$227$	143145.	0.184379	0.0921897	+	0.995741i	$0.470613\pi$
$228$	351547.	0.447864
$229$	−1.45047e6	−1.82776	−0.913882	−	0.405980i	$-0.866930\pi$
$229$	−1.45047e6	−1.82776	−0.913882	+	0.405980i	$0.866930\pi$
$230$	584035.	0.727979
$231$	434982.	0.536342
$232$	845582.	1.03142
$233$	17913.6	0.0216169	0.0108085	−	0.999942i	$-0.496559\pi$
$233$	17913.6	0.0216169	0.0108085	+	0.999942i	$0.496559\pi$
$234$	95790.4	0.114362
$235$	−255117.	−0.301349
$236$	121552.	0.142063
$237$	−923230.	−1.06768
$238$	366106.	0.418953
$239$	398359.	0.451108	0.225554	−	0.974231i	$-0.427581\pi$
$239$	398359.	0.451108	0.225554	+	0.974231i	$0.427581\pi$
$240$	287790.	0.322513
$241$	−1.15213e6	−1.27778	−0.638892	−	0.769296i	$-0.720607\pi$
$241$	−1.15213e6	−1.27778	−0.638892	+	0.769296i	$0.720607\pi$
$242$	2.51888e6	2.76483
$243$	−59049.0	−0.0641500
$244$	−692177.	−0.744290
$245$	−308089.	−0.327915
$246$	634515.	0.668504
$247$	−389073.	−0.405778
$248$	480694.	0.496294
$249$	−819568.	−0.837696
$250$	109338.	0.110642
$251$	−800083.	−0.801587	−0.400794	−	0.916168i	$-0.631266\pi$
$251$	−800083.	−0.801587	−0.400794	+	0.916168i	$0.631266\pi$
$252$	92021.0	0.0912821
$253$	−2.40975e6	−2.36685
$254$	−2.06246e6	−2.00586
$255$	−175806.	−0.169311
$256$	1.28188e6	1.22250
$257$	−669863.	−0.632635	−0.316317	−	0.948653i	$-0.602446\pi$
$257$	−669863.	−0.632635	−0.316317	+	0.948653i	$0.602446\pi$
$258$	−750358.	−0.701810
$259$	−833621.	−0.772181
$260$	71684.2	0.0657643
$261$	−651082.	−0.591608
$262$	−446755.	−0.402084
$263$	−493932.	−0.440329	−0.220165	−	0.975463i	$-0.570659\pi$
$263$	−493932.	−0.440329	−0.220165	+	0.975463i	$0.570659\pi$
$264$	−683395.	−0.603479
$265$	−341401.	−0.298641
$266$	−1.07870e6	−0.934749
$267$	−184867.	−0.158702
$268$	495649.	0.421538
$269$	2.15725e6	1.81769	0.908843	−	0.417139i	$-0.136967\pi$
$269$	2.15725e6	1.81769	0.908843	+	0.417139i	$0.136967\pi$
$270$	−127532.	−0.106465
$271$	1.46880e6	1.21489	0.607447	−	0.794360i	$-0.292194\pi$
$271$	1.46880e6	1.21489	0.607447	+	0.794360i	$0.292194\pi$
$272$	−999413.	−0.819073
$273$	−101844.	−0.0827042
$274$	1.19233e6	0.959447
$275$	−451133.	−0.359727
$276$	−509786.	−0.402824
$277$	−2.54437e6	−1.99242	−0.996211	−	0.0869688i	$-0.972282\pi$
$277$	−2.54437e6	−1.99242	−0.996211	+	0.0869688i	$0.972282\pi$
$278$	−2.60535e6	−2.02187
$279$	−370125.	−0.284668
$280$	−176096.	−0.134232
$281$	−541195.	−0.408873	−0.204436	−	0.978880i	$-0.565536\pi$
$281$	−541195.	−0.408873	−0.204436	+	0.978880i	$0.565536\pi$
$282$	642676.	0.481249
$283$	747106.	0.554518	0.277259	−	0.960795i	$-0.410574\pi$
$283$	747106.	0.554518	0.277259	+	0.960795i	$0.410574\pi$
$284$	1.20774e6	0.888542
$285$	517996.	0.377759
$286$	−853614.	−0.617087
$287$	−674611.	−0.483447
$288$	−452312.	−0.321334
$289$	−809330.	−0.570008
$290$	−1.40618e6	−0.981852
$291$	−342852.	−0.237342
$292$	633021.	0.434472
$293$	587050.	0.399490	0.199745	−	0.979848i	$-0.435989\pi$
$293$	587050.	0.399490	0.199745	+	0.979848i	$0.435989\pi$
$294$	776122.	0.523674
$295$	179104.	0.119826
$296$	1.30969e6	0.868839
$297$	526201.	0.346147
$298$	−3.63232e6	−2.36943
$299$	564202.	0.364970
$300$	−95437.6	−0.0612232
$301$	797775.	0.507533
$302$	777972.	0.490848
$303$	455469.	0.285005
$304$	2.94467e6	1.82748
$305$	−1.01991e6	−0.627784
$306$	442882.	0.270386
$307$	−835807.	−0.506128	−0.253064	−	0.967450i	$-0.581438\pi$
$307$	−835807.	−0.506128	−0.253064	+	0.967450i	$0.581438\pi$
$308$	−820023.	−0.492549
$309$	563862.	0.335952
$310$	−799381.	−0.472443
$311$	−1.06297e6	−0.623192	−0.311596	−	0.950215i	$-0.600864\pi$
$311$	−1.06297e6	−0.623192	−0.311596	+	0.950215i	$0.600864\pi$
$312$	160005.	0.0930567
$313$	−1.54176e6	−0.889520	−0.444760	−	0.895650i	$-0.646711\pi$
$313$	−1.54176e6	−0.889520	−0.444760	+	0.895650i	$0.646711\pi$
$314$	−1.81597e6	−1.03940
$315$	135591.	0.0769934
$316$	1.74046e6	0.980498
$317$	−424457.	−0.237239	−0.118619	−	0.992940i	$-0.537847\pi$
$317$	−424457.	−0.237239	−0.118619	+	0.992940i	$0.537847\pi$
$318$	860038.	0.476925
$319$	5.80196e6	3.19226
$320$	46367.8	0.0253129
$321$	−482288.	−0.261243
$322$	1.56424e6	0.840744
$323$	−1.79886e6	−0.959379
$324$	111318.	0.0589121
$325$	105625.	0.0554700
$326$	1.56835e6	0.817334
$327$	184432.	0.0953819
$328$	1.05987e6	0.543963
$329$	−683289.	−0.348028
$330$	1.13647e6	0.574477
$331$	1.46376e6	0.734342	0.367171	−	0.930153i	$-0.380326\pi$
$331$	1.46376e6	0.734342	0.367171	+	0.930153i	$0.380326\pi$
$332$	1.54504e6	0.769297
$333$	−1.00844e6	−0.498354
$334$	4.22726e6	2.07345
$335$	730327.	0.355554
$336$	770797.	0.372470
$337$	−2.50149e6	−1.19984	−0.599921	−	0.800059i	$-0.704802\pi$
$337$	−2.50149e6	−1.19984	−0.599921	+	0.800059i	$0.704802\pi$
$338$	199859.	0.0951551
$339$	946894.	0.447509
$340$	331428.	0.155486
$341$	3.29828e6	1.53604
$342$	−1.30491e6	−0.603273
$343$	−1.95054e6	−0.895197
$344$	−1.25337e6	−0.571064
$345$	−751157.	−0.339768
$346$	1.22265e6	0.549049
$347$	1.71568e6	0.764915	0.382457	−	0.923973i	$-0.375078\pi$
$347$	1.71568e6	0.764915	0.382457	+	0.923973i	$0.375078\pi$
$348$	1.22741e6	0.543303
$349$	3.12794e6	1.37466	0.687329	−	0.726346i	$-0.258783\pi$
$349$	3.12794e6	1.37466	0.687329	+	0.726346i	$0.258783\pi$
$350$	292843.	0.127781
$351$	−123201.	−0.0533761
$352$	4.03067e6	1.73389
$353$	−1.83043e6	−0.781835	−0.390918	−	0.920426i	$-0.627842\pi$
$353$	−1.83043e6	−0.781835	−0.390918	+	0.920426i	$0.627842\pi$
$354$	−451189.	−0.191359
$355$	1.77958e6	0.749455
$356$	348510.	0.145744
$357$	−470868.	−0.195537
$358$	−4.30079e6	−1.77354
$359$	−2.29833e6	−0.941190	−0.470595	−	0.882349i	$-0.655961\pi$
$359$	−2.29833e6	−0.941190	−0.470595	+	0.882349i	$0.655961\pi$
$360$	−213025.	−0.0866311
$361$	2.82405e6	1.14052
$362$	3.98851e6	1.59970
$363$	−3.23966e6	−1.29043
$364$	191994.	0.0759513
$365$	932742.	0.366462
$366$	2.56929e6	1.00256
$367$	−4.58322e6	−1.77626	−0.888128	−	0.459597i	$-0.847994\pi$
$367$	−4.58322e6	−1.77626	−0.888128	+	0.459597i	$0.847994\pi$
$368$	−4.27013e6	−1.64370
$369$	−816082.	−0.312010
$370$	−2.17798e6	−0.827084
$371$	−914386.	−0.344901
$372$	697755.	0.261424
$373$	494132.	0.183896	0.0919478	−	0.995764i	$-0.470691\pi$
$373$	494132.	0.183896	0.0919478	+	0.995764i	$0.470691\pi$
$374$	−3.94664e6	−1.45898
$375$	−140625.	−0.0516398
$376$	1.07351e6	0.391593
$377$	−1.35843e6	−0.492248
$378$	−341572.	−0.122957
$379$	2.51002e6	0.897591	0.448795	−	0.893635i	$-0.351853\pi$
$379$	2.51002e6	0.897591	0.448795	+	0.893635i	$0.351853\pi$
$380$	−976520.	−0.346914
$381$	2.65264e6	0.936193
$382$	1.77836e6	0.623535
$383$	1.69626e6	0.590876	0.295438	−	0.955362i	$-0.404534\pi$
$383$	1.69626e6	0.590876	0.295438	+	0.955362i	$0.404534\pi$
$384$	−1.72503e6	−0.596991
$385$	−1.20828e6	−0.415449
$386$	193242.	0.0660134
$387$	965075.	0.327554
$388$	646340.	0.217962
$389$	−299373.	−0.100309	−0.0501543	−	0.998741i	$-0.515971\pi$
$389$	−299373.	−0.100309	−0.0501543	+	0.998741i	$0.515971\pi$
$390$	−266084.	−0.0885846
$391$	2.60856e6	0.862896
$392$	1.29641e6	0.426115
$393$	574595.	0.187664
$394$	6.18441e6	2.00705
$395$	2.56453e6	0.827018
$396$	−991988.	−0.317884
$397$	−1.40190e6	−0.446418	−0.223209	−	0.974771i	$-0.571653\pi$
$397$	−1.40190e6	−0.446418	−0.223209	+	0.974771i	$0.571653\pi$
$398$	2.62158e6	0.829573
$399$	1.38737e6	0.436274
$400$	−799416.	−0.249817
$401$	2.32824e6	0.723049	0.361524	−	0.932363i	$-0.382256\pi$
$401$	2.32824e6	0.723049	0.361524	+	0.932363i	$0.382256\pi$
$402$	−1.83980e6	−0.567812
$403$	−772236.	−0.236858
$404$	−858645.	−0.261734
$405$	164025.	0.0496904
$406$	−3.76622e6	−1.13394
$407$	8.98645e6	2.68907
$408$	739775.	0.220014
$409$	625884.	0.185006	0.0925029	−	0.995712i	$-0.470513\pi$
$409$	625884.	0.185006	0.0925029	+	0.995712i	$0.470513\pi$
$410$	−1.76254e6	−0.517821
$411$	−1.53352e6	−0.447801
$412$	−1.06299e6	−0.308521
$413$	479700.	0.138387
$414$	1.89227e6	0.542604
$415$	2.27658e6	0.648877
$416$	−943712.	−0.267366
$417$	3.35087e6	0.943665
$418$	1.16284e7	3.25520
$419$	−1.38874e6	−0.386442	−0.193221	−	0.981155i	$-0.561893\pi$
$419$	−1.38874e6	−0.386442	−0.193221	+	0.981155i	$0.561893\pi$
$420$	−255614.	−0.0707068
$421$	−6.45976e6	−1.77628	−0.888139	−	0.459575i	$-0.848002\pi$
$421$	−6.45976e6	−1.77628	−0.888139	+	0.459575i	$0.848002\pi$
$422$	−7.99358e6	−2.18504
$423$	−826579.	−0.224612
$424$	1.43658e6	0.388075
$425$	488351.	0.131147
$426$	−4.48301e6	−1.19687
$427$	−2.73165e6	−0.725029
$428$	909204.	0.239912
$429$	1.09788e6	0.288012
$430$	2.08433e6	0.543620
$431$	−3.08684e6	−0.800426	−0.400213	−	0.916422i	$-0.631064\pi$
$431$	−3.08684e6	−0.800426	−0.400213	+	0.916422i	$0.631064\pi$
$432$	932439.	0.240387
$433$	−4.59588e6	−1.17801	−0.589005	−	0.808130i	$-0.700480\pi$
$433$	−4.59588e6	−1.17801	−0.589005	+	0.808130i	$0.700480\pi$
$434$	−2.14101e6	−0.545625
$435$	1.80856e6	0.458258
$436$	−347688.	−0.0875938
$437$	−7.68586e6	−1.92526
$438$	−2.34971e6	−0.585233
$439$	−2.84441e6	−0.704420	−0.352210	−	0.935921i	$-0.614570\pi$
$439$	−2.84441e6	−0.704420	−0.352210	+	0.935921i	$0.614570\pi$
$440$	1.89832e6	0.467453
$441$	−998210.	−0.244414
$442$	924037.	0.224975
$443$	4.20072e6	1.01698	0.508492	−	0.861067i	$-0.330203\pi$
$443$	4.20072e6	1.01698	0.508492	+	0.861067i	$0.330203\pi$
$444$	1.90109e6	0.457663
$445$	513521.	0.122930
$446$	−207366.	−0.0493629
$447$	4.67172e6	1.10588
$448$	124188.	0.0292338
$449$	−2.79787e6	−0.654955	−0.327478	−	0.944859i	$-0.606199\pi$
$449$	−2.79787e6	−0.654955	−0.327478	+	0.944859i	$0.606199\pi$
$450$	354255.	0.0824677
$451$	7.27232e6	1.68357
$452$	−1.78507e6	−0.410970
$453$	−1.00059e6	−0.229092
$454$	1.00168e6	0.228080
$455$	282899.	0.0640624
$456$	−2.17967e6	−0.490885
$457$	2.61344e6	0.585358	0.292679	−	0.956211i	$-0.405453\pi$
$457$	2.61344e6	0.585358	0.292679	+	0.956211i	$0.405453\pi$
$458$	−1.01498e7	−2.26097
$459$	−569613.	−0.126197
$460$	1.41607e6	0.312026
$461$	5.12210e6	1.12253	0.561263	−	0.827638i	$-0.310316\pi$
$461$	5.12210e6	1.12253	0.561263	+	0.827638i	$0.310316\pi$
$462$	3.04384e6	0.663464
$463$	5.44971e6	1.18146	0.590732	−	0.806868i	$-0.298839\pi$
$463$	5.44971e6	1.18146	0.590732	+	0.806868i	$0.298839\pi$
$464$	1.02812e7	2.21691
$465$	1.02813e6	0.220503
$466$	125353.	0.0267405
$467$	1.43759e6	0.305031	0.152516	−	0.988301i	$-0.451263\pi$
$467$	1.43759e6	0.305031	0.152516	+	0.988301i	$0.451263\pi$
$468$	232257.	0.0490178
$469$	1.95606e6	0.410629
$470$	−1.78521e6	−0.372774
$471$	2.33561e6	0.485118
$472$	−753651.	−0.155710
$473$	−8.60004e6	−1.76745
$474$	−6.46042e6	−1.32073
$475$	−1.43888e6	−0.292611
$476$	887675.	0.179571
$477$	−1.10614e6	−0.222594
$478$	2.78757e6	0.558027
$479$	−5.58674e6	−1.11255	−0.556276	−	0.830998i	$-0.687770\pi$
$479$	−5.58674e6	−1.11255	−0.556276	+	0.830998i	$0.687770\pi$
$480$	1.25642e6	0.248904
$481$	−2.10402e6	−0.414656
$482$	−8.06215e6	−1.58064
$483$	−2.01185e6	−0.392399
$484$	6.10737e6	1.18506
$485$	952366.	0.183844
$486$	−413202.	−0.0793546
$487$	−4.65236e6	−0.888895	−0.444448	−	0.895805i	$-0.646600\pi$
$487$	−4.65236e6	−0.888895	−0.444448	+	0.895805i	$0.646600\pi$
$488$	4.29166e6	0.815785
$489$	−2.01714e6	−0.381473
$490$	−2.15589e6	−0.405636
$491$	−483556.	−0.0905197	−0.0452598	−	0.998975i	$-0.514412\pi$
$491$	−483556.	−0.0905197	−0.0452598	+	0.998975i	$0.514412\pi$
$492$	1.53847e6	0.286534
$493$	−6.28062e6	−1.16382
$494$	−2.72258e6	−0.501954
$495$	−1.46167e6	−0.268124
$496$	5.84462e6	1.06672
$497$	4.76630e6	0.865546
$498$	−5.73502e6	−1.03624
$499$	−7.51098e6	−1.35035	−0.675173	−	0.737659i	$-0.735931\pi$
$499$	−7.51098e6	−1.35035	−0.675173	+	0.737659i	$0.735931\pi$
$500$	265104.	0.0474233
$501$	−5.43689e6	−0.967735
$502$	−5.59868e6	−0.991576
$503$	5.82616e6	1.02675	0.513373	−	0.858166i	$-0.328396\pi$
$503$	5.82616e6	1.02675	0.513373	+	0.858166i	$0.328396\pi$
$504$	−570552.	−0.100050
$505$	−1.26519e6	−0.220764
$506$	−1.68625e7	−2.92784
$507$	−257049.	−0.0444116
$508$	−5.00072e6	−0.859752
$509$	−406063.	−0.0694703	−0.0347351	−	0.999397i	$-0.511059\pi$
$509$	−406063.	−0.0694703	−0.0347351	+	0.999397i	$0.511059\pi$
$510$	−1.23023e6	−0.209440
$511$	2.49819e6	0.423228
$512$	2.83668e6	0.478229
$513$	1.67831e6	0.281565
$514$	−4.68744e6	−0.782579
$515$	−1.56628e6	−0.260227
$516$	−1.81935e6	−0.300809
$517$	7.36587e6	1.21199
$518$	−5.83336e6	−0.955200
$519$	−1.57251e6	−0.256257
$520$	−444459.	−0.0720814
$521$	1.00945e7	1.62926	0.814630	−	0.579981i	$-0.196940\pi$
$521$	1.00945e7	1.62926	0.814630	+	0.579981i	$0.196940\pi$
$522$	−4.55602e6	−0.731829
$523$	2.37804e6	0.380158	0.190079	−	0.981769i	$-0.439126\pi$
$523$	2.37804e6	0.380158	0.190079	+	0.981769i	$0.439126\pi$
$524$	−1.08322e6	−0.172341
$525$	−376641.	−0.0596388
$526$	−3.45635e6	−0.544695
$527$	−3.57039e6	−0.560002
$528$	−8.30921e6	−1.29710
$529$	4.70908e6	0.731639
$530$	−2.38899e6	−0.369424
$531$	580297.	0.0893129
$532$	−2.61545e6	−0.400651
$533$	−1.70269e6	−0.259608
$534$	−1.29363e6	−0.196317
$535$	1.33969e6	0.202358
$536$	−3.07314e6	−0.462030
$537$	5.53147e6	0.827761
$538$	1.50956e7	2.24851
$539$	8.89531e6	1.31883
$540$	−309218.	−0.0456331
$541$	2.95079e6	0.433456	0.216728	−	0.976232i	$-0.430462\pi$
$541$	2.95079e6	0.433456	0.216728	+	0.976232i	$0.430462\pi$
$542$	1.02781e7	1.50284
$543$	−5.12983e6	−0.746626
$544$	−4.36320e6	−0.632132
$545$	−512310.	−0.0738825
$546$	−712663.	−0.102306
$547$	−1.11440e7	−1.59248	−0.796240	−	0.604981i	$-0.793181\pi$
$547$	−1.11440e7	−1.59248	−0.796240	+	0.604981i	$0.793181\pi$
$548$	2.89097e6	0.411238
$549$	−3.30449e6	−0.467923
$550$	−3.15686e6	−0.444988
$551$	1.85052e7	2.59666
$552$	3.16079e6	0.441518
$553$	6.86867e6	0.955123
$554$	−1.78046e7	−2.46466
$555$	2.80121e6	0.386023
$556$	−6.31703e6	−0.866614
$557$	−1.82804e6	−0.249659	−0.124830	−	0.992178i	$-0.539838\pi$
$557$	−1.82804e6	−0.249659	−0.124830	+	0.992178i	$0.539838\pi$
$558$	−2.58999e6	−0.352138
$559$	2.01355e6	0.272542
$560$	−2.14110e6	−0.288514
$561$	5.07597e6	0.680945
$562$	−3.78708e6	−0.505782
$563$	7.18745e6	0.955661	0.477830	−	0.878452i	$-0.341423\pi$
$563$	7.18745e6	0.955661	0.477830	+	0.878452i	$0.341423\pi$
$564$	1.55826e6	0.206272
$565$	−2.63026e6	−0.346639
$566$	5.22796e6	0.685948
$567$	439314.	0.0573875
$568$	−7.48827e6	−0.973892
$569$	9.09907e6	1.17819	0.589096	−	0.808063i	$-0.299484\pi$
$569$	9.09907e6	1.17819	0.589096	+	0.808063i	$0.299484\pi$
$570$	3.62474e6	0.467294
$571$	−7.53787e6	−0.967517	−0.483758	−	0.875202i	$-0.660729\pi$
$571$	−7.53787e6	−0.967517	−0.483758	+	0.875202i	$0.660729\pi$
$572$	−2.06970e6	−0.264495
$573$	−2.28724e6	−0.291021
$574$	−4.72067e6	−0.598032
$575$	2.08655e6	0.263184
$576$	150232.	0.0188671
$577$	297945.	0.0372560	0.0186280	−	0.999826i	$-0.494070\pi$
$577$	297945.	0.0372560	0.0186280	+	0.999826i	$0.494070\pi$
$578$	−5.66339e6	−0.705110
$579$	−248538.	−0.0308103
$580$	−3.40947e6	−0.420840
$581$	6.09743e6	0.749388
$582$	−2.39915e6	−0.293595
$583$	9.85710e6	1.20110
$584$	−3.92488e6	−0.476206
$585$	342225.	0.0413449
$586$	4.10795e6	0.494176
$587$	1.07712e7	1.29024	0.645120	−	0.764081i	$-0.276807\pi$
$587$	1.07712e7	1.29024	0.645120	+	0.764081i	$0.276807\pi$
$588$	1.88181e6	0.224457
$589$	1.05198e7	1.24945
$590$	1.25330e6	0.148226
$591$	−7.95410e6	−0.936747
$592$	1.59242e7	1.86746
$593$	437753.	0.0511202	0.0255601	−	0.999673i	$-0.491863\pi$
$593$	437753.	0.0511202	0.0255601	+	0.999673i	$0.491863\pi$
$594$	3.68216e6	0.428190
$595$	1.30797e6	0.151462
$596$	−8.80707e6	−1.01558
$597$	−3.37174e6	−0.387185
$598$	3.94807e6	0.451474
$599$	−1.62933e7	−1.85542	−0.927708	−	0.373306i	$-0.878224\pi$
$599$	−1.62933e7	−1.85542	−0.927708	+	0.373306i	$0.878224\pi$
$600$	591736.	0.0671042
$601$	−9.07820e6	−1.02521	−0.512606	−	0.858624i	$-0.671320\pi$
$601$	−9.07820e6	−1.02521	−0.512606	+	0.858624i	$0.671320\pi$
$602$	5.58253e6	0.627827
$603$	2.36626e6	0.265014
$604$	1.88630e6	0.210387
$605$	8.99906e6	0.999559
$606$	3.18720e6	0.352556
$607$	−1.40774e6	−0.155078	−0.0775389	−	0.996989i	$-0.524706\pi$
$607$	−1.40774e6	−0.155078	−0.0775389	+	0.996989i	$0.524706\pi$
$608$	1.28557e7	1.41039
$609$	4.84393e6	0.529242
$610$	−7.13691e6	−0.776579
$611$	−1.72459e6	−0.186889
$612$	1.07383e6	0.115893
$613$	−7.66603e6	−0.823985	−0.411992	−	0.911187i	$-0.635167\pi$
$613$	−7.66603e6	−0.823985	−0.411992	+	0.911187i	$0.635167\pi$
$614$	−5.84866e6	−0.626088
$615$	2.26689e6	0.241682
$616$	5.08434e6	0.539862
$617$	8.80623e6	0.931273	0.465636	−	0.884976i	$-0.345825\pi$
$617$	8.80623e6	0.931273	0.465636	+	0.884976i	$0.345825\pi$
$618$	3.94569e6	0.415578
$619$	1.40197e6	0.147066	0.0735331	−	0.997293i	$-0.476573\pi$
$619$	1.40197e6	0.147066	0.0735331	+	0.997293i	$0.476573\pi$
$620$	−1.93821e6	−0.202498
$621$	−2.43375e6	−0.253248
$622$	−7.43829e6	−0.770899
$623$	1.37538e6	0.141972
$624$	1.94546e6	0.200014
$625$	390625.	0.0400000
$626$	−1.07886e7	−1.10035
$627$	−1.49558e7	−1.51930
$628$	−4.40306e6	−0.445508
$629$	−9.72783e6	−0.980368
$630$	948812.	0.0952421
$631$	−9.84323e6	−0.984157	−0.492078	−	0.870551i	$-0.663763\pi$
$631$	−9.84323e6	−0.984157	−0.492078	+	0.870551i	$0.663763\pi$
$632$	−1.07913e7	−1.07468
$633$	1.02809e7	1.01982
$634$	−2.97019e6	−0.293468
$635$	−7.36844e6	−0.725172
$636$	2.08528e6	0.204419
$637$	−2.08268e6	−0.203364
$638$	4.05999e7	3.94888
$639$	5.76583e6	0.558611
$640$	4.79174e6	0.462427
$641$	−1.74155e7	−1.67414	−0.837070	−	0.547096i	$-0.815733\pi$
$641$	−1.74155e7	−1.67414	−0.837070	+	0.547096i	$0.815733\pi$
$642$	−3.37487e6	−0.323161
$643$	1.23961e7	1.18238	0.591190	−	0.806533i	$-0.298659\pi$
$643$	1.23961e7	1.18238	0.591190	+	0.806533i	$0.298659\pi$
$644$	3.79271e6	0.360359
$645$	−2.68076e6	−0.253723
$646$	−1.25877e7	−1.18677
$647$	−1.89320e7	−1.77802	−0.889010	−	0.457887i	$-0.848606\pi$
$647$	−1.89320e7	−1.77802	−0.889010	+	0.457887i	$0.848606\pi$
$648$	−690200.	−0.0645710
$649$	−5.17118e6	−0.481923
$650$	739124.	0.0686173
$651$	2.75366e6	0.254659
$652$	3.80268e6	0.350325
$653$	1.33112e7	1.22161	0.610806	−	0.791780i	$-0.290845\pi$
$653$	1.33112e7	1.22161	0.610806	+	0.791780i	$0.290845\pi$
$654$	1.29058e6	0.117989
$655$	−1.59610e6	−0.145364
$656$	1.28867e7	1.16918
$657$	3.02208e6	0.273145
$658$	−4.78139e6	−0.430517
$659$	−6.04189e6	−0.541950	−0.270975	−	0.962586i	$-0.587346\pi$
$659$	−6.04189e6	−0.541950	−0.270975	+	0.962586i	$0.587346\pi$
$660$	2.75552e6	0.246232
$661$	6.74321e6	0.600293	0.300146	−	0.953893i	$-0.402964\pi$
$661$	6.74321e6	0.600293	0.300146	+	0.953893i	$0.402964\pi$
$662$	1.02428e7	0.908393
$663$	−1.18845e6	−0.105002
$664$	−9.57960e6	−0.843193
$665$	−3.85380e6	−0.337936
$666$	−7.05666e6	−0.616472
$667$	−2.68348e7	−2.33552
$668$	1.02496e7	0.888718
$669$	266705.	0.0230391
$670$	5.11055e6	0.439826
$671$	2.94472e7	2.52487
$672$	3.36512e6	0.287460
$673$	−9.59924e6	−0.816957	−0.408478	−	0.912768i	$-0.633940\pi$
$673$	−9.59924e6	−0.816957	−0.408478	+	0.912768i	$0.633940\pi$
$674$	−1.75045e7	−1.48423
$675$	−455625.	−0.0384900
$676$	484585.	0.0407853
$677$	3.66549e6	0.307369	0.153685	−	0.988120i	$-0.450886\pi$
$677$	3.66549e6	0.307369	0.153685	+	0.988120i	$0.450886\pi$
$678$	6.62600e6	0.553576
$679$	2.55075e6	0.212322
$680$	−2.05493e6	−0.170422
$681$	−1.28831e6	−0.106451
$682$	2.30801e7	1.90010
$683$	9.53070e6	0.781759	0.390879	−	0.920442i	$-0.372171\pi$
$683$	9.53070e6	0.781759	0.390879	+	0.920442i	$0.372171\pi$
$684$	−3.16392e6	−0.258575
$685$	4.25978e6	0.346865
$686$	−1.36491e7	−1.10737
$687$	1.30542e7	1.05526
$688$	−1.52394e7	−1.22743
$689$	−2.30787e6	−0.185210
$690$	−5.25631e6	−0.420299
$691$	9.67074e6	0.770486	0.385243	−	0.922815i	$-0.374118\pi$
$691$	9.67074e6	0.770486	0.385243	+	0.922815i	$0.374118\pi$
$692$	2.96448e6	0.235333
$693$	−3.91484e6	−0.309657
$694$	1.20057e7	0.946212
$695$	−9.30798e6	−0.730960
$696$	−7.61024e6	−0.595491
$697$	−7.87229e6	−0.613789
$698$	2.18881e7	1.70047
$699$	−161223.	−0.0124805
$700$	710038.	0.0547692
$701$	1.03870e7	0.798356	0.399178	−	0.916873i	$-0.369295\pi$
$701$	1.03870e7	0.798356	0.399178	+	0.916873i	$0.369295\pi$
$702$	−862114.	−0.0660270
$703$	2.86621e7	2.18736
$704$	−1.33875e6	−0.101805
$705$	2.29605e6	0.173984
$706$	−1.28086e7	−0.967143
$707$	−3.38861e6	−0.254960
$708$	−1.09397e6	−0.0820203
$709$	3.75669e6	0.280666	0.140333	−	0.990104i	$-0.455183\pi$
$709$	3.75669e6	0.280666	0.140333	+	0.990104i	$0.455183\pi$
$710$	1.24528e7	0.927088
$711$	8.30907e6	0.616423
$712$	−2.16084e6	−0.159743
$713$	−1.52550e7	−1.12380
$714$	−3.29496e6	−0.241882
$715$	−3.04966e6	−0.223093
$716$	−1.04279e7	−0.760173
$717$	−3.58523e6	−0.260447
$718$	−1.60829e7	−1.16427
$719$	161631.	0.0116601	0.00583006	−	0.999983i	$-0.498144\pi$
$719$	161631.	0.0116601	0.00583006	+	0.999983i	$0.498144\pi$
$720$	−2.59011e6	−0.186203
$721$	−4.19503e6	−0.300536
$722$	1.97616e7	1.41085
$723$	1.03691e7	0.737729
$724$	9.67068e6	0.685663
$725$	−5.02378e6	−0.354965
$726$	−2.26699e7	−1.59628
$727$	−1.05397e7	−0.739594	−0.369797	−	0.929112i	$-0.620573\pi$
$727$	−1.05397e7	−0.739594	−0.369797	+	0.929112i	$0.620573\pi$
$728$	−1.19041e6	−0.0832469
$729$	531441.	0.0370370
$730$	6.52697e6	0.453320
$731$	9.30954e6	0.644369
$732$	6.22959e6	0.429716
$733$	1.40432e7	0.965400	0.482700	−	0.875786i	$-0.339656\pi$
$733$	1.40432e7	0.965400	0.482700	+	0.875786i	$0.339656\pi$
$734$	−3.20716e7	−2.19726
$735$	2.77281e6	0.189322
$736$	−1.86424e7	−1.26855
$737$	−2.10864e7	−1.42999
$738$	−5.71063e6	−0.385961
$739$	−1.13745e6	−0.0766165	−0.0383083	−	0.999266i	$-0.512197\pi$
$739$	−1.13745e6	−0.0766165	−0.0383083	+	0.999266i	$0.512197\pi$
$740$	−5.28081e6	−0.354504
$741$	3.50165e6	0.234276
$742$	−6.39852e6	−0.426648
$743$	2.74898e6	0.182684	0.0913419	−	0.995820i	$-0.470884\pi$
$743$	2.74898e6	0.182684	0.0913419	+	0.995820i	$0.470884\pi$
$744$	−4.32625e6	−0.286536
$745$	−1.29770e7	−0.856611
$746$	3.45775e6	0.227482
$747$	7.37611e6	0.483644
$748$	−9.56916e6	−0.625345
$749$	3.58814e6	0.233703
$750$	−984040.	−0.0638792
$751$	2.22457e6	0.143928	0.0719642	−	0.997407i	$-0.477073\pi$
$751$	2.22457e6	0.143928	0.0719642	+	0.997407i	$0.477073\pi$
$752$	1.30525e7	0.841681
$753$	7.20075e6	0.462797
$754$	−9.50577e6	−0.608918
$755$	2.77942e6	0.177454
$756$	−828189.	−0.0527017
$757$	−1.01829e7	−0.645849	−0.322924	−	0.946425i	$-0.604666\pi$
$757$	−1.01829e7	−0.645849	−0.322924	+	0.946425i	$0.604666\pi$
$758$	1.75641e7	1.11033
$759$	2.16878e7	1.36650
$760$	6.05465e6	0.380238
$761$	7.31991e6	0.458189	0.229094	−	0.973404i	$-0.426424\pi$
$761$	7.31991e6	0.458189	0.229094	+	0.973404i	$0.426424\pi$
$762$	1.85622e7	1.15809
$763$	−1.37214e6	−0.0853269
$764$	4.31187e6	0.267259
$765$	1.58226e6	0.0977516
$766$	1.18698e7	0.730924
$767$	1.21074e6	0.0743128
$768$	−1.15369e7	−0.705809
$769$	739800.	0.0451126	0.0225563	−	0.999746i	$-0.492819\pi$
$769$	739800.	0.0451126	0.0225563	+	0.999746i	$0.492819\pi$
$770$	−8.45512e6	−0.513917
$771$	6.02876e6	0.365252
$772$	468540.	0.0282946
$773$	2.62668e7	1.58110	0.790549	−	0.612399i	$-0.209795\pi$
$773$	2.62668e7	1.58110	0.790549	+	0.612399i	$0.209795\pi$
$774$	6.75322e6	0.405190
$775$	−2.85590e6	−0.170801
$776$	−4.00746e6	−0.238899
$777$	7.50259e6	0.445819
$778$	−2.09489e6	−0.124083
$779$	2.31949e7	1.36946
$780$	−645158.	−0.0379690
$781$	−5.13808e7	−3.01421
$782$	1.82537e7	1.06742
$783$	5.85974e6	0.341565
$784$	1.57627e7	0.915882
$785$	−6.48780e6	−0.375771
$786$	4.02080e6	0.232143
$787$	−216388.	−0.0124537	−0.00622683	−	0.999981i	$-0.501982\pi$
$787$	−216388.	−0.0124537	−0.00622683	+	0.999981i	$0.501982\pi$
$788$	1.49950e7	0.860260
$789$	4.44539e6	0.254224
$790$	1.79456e7	1.02303
$791$	−7.04472e6	−0.400334
$792$	6.15056e6	0.348419
$793$	−6.89456e6	−0.389335
$794$	−9.80998e6	−0.552226
$795$	3.07261e6	0.172421
$796$	6.35637e6	0.355571
$797$	−2.35778e7	−1.31479	−0.657395	−	0.753546i	$-0.728342\pi$
$797$	−2.35778e7	−1.31479	−0.657395	+	0.753546i	$0.728342\pi$
$798$	9.70827e6	0.539678
$799$	−7.97355e6	−0.441860
$800$	−3.49006e6	−0.192800
$801$	1.66381e6	0.0916266
$802$	1.62922e7	0.894423
$803$	−2.69306e7	−1.47386
$804$	−4.46084e6	−0.243375
$805$	5.58847e6	0.303951
$806$	−5.40382e6	−0.292997
$807$	−1.94152e7	−1.04944
$808$	5.32380e6	0.286875
$809$	−2.22061e7	−1.19289	−0.596445	−	0.802654i	$-0.703421\pi$
$809$	−2.22061e7	−1.19289	−0.596445	+	0.802654i	$0.703421\pi$
$810$	1.14778e6	0.0614678
$811$	−2.16203e7	−1.15428	−0.577138	−	0.816646i	$-0.695831\pi$
$811$	−2.16203e7	−1.15428	−0.577138	+	0.816646i	$0.695831\pi$
$812$	−9.13171e6	−0.486029
$813$	−1.32192e7	−0.701419
$814$	6.28838e7	3.32642
$815$	5.60316e6	0.295487
$816$	8.99472e6	0.472892
$817$	−2.74296e7	−1.43769
$818$	4.37970e6	0.228855
$819$	916593.	0.0477493
$820$	−4.27352e6	−0.221948
$821$	2.37570e6	0.123008	0.0615039	−	0.998107i	$-0.480410\pi$
$821$	2.37570e6	0.123008	0.0615039	+	0.998107i	$0.480410\pi$
$822$	−1.07310e7	−0.553937
$823$	8.01991e6	0.412733	0.206367	−	0.978475i	$-0.433836\pi$
$823$	8.01991e6	0.412733	0.206367	+	0.978475i	$0.433836\pi$
$824$	6.59076e6	0.338156
$825$	4.06020e6	0.207688
$826$	3.35676e6	0.171187
$827$	2.57093e6	0.130715	0.0653576	−	0.997862i	$-0.479181\pi$
$827$	2.57093e6	0.130715	0.0653576	+	0.997862i	$0.479181\pi$
$828$	4.58807e6	0.232570
$829$	1.48897e7	0.752488	0.376244	−	0.926521i	$-0.377215\pi$
$829$	1.48897e7	0.752488	0.376244	+	0.926521i	$0.377215\pi$
$830$	1.59306e7	0.802671
$831$	2.28994e7	1.15033
$832$	313446.	0.0156984
$833$	−9.62918e6	−0.480813
$834$	2.34481e7	1.16733
$835$	1.51025e7	0.749605
$836$	2.81946e7	1.39524
$837$	3.33113e6	0.164353
$838$	−9.71784e6	−0.478035
$839$	2.00964e7	0.985628	0.492814	−	0.870135i	$-0.335968\pi$
$839$	2.00964e7	0.985628	0.492814	+	0.870135i	$0.335968\pi$
$840$	1.58487e6	0.0774987
$841$	4.40990e7	2.15000
$842$	−4.52029e7	−2.19728
$843$	4.87075e6	0.236063
$844$	−1.93815e7	−0.936551
$845$	714025.	0.0344010
$846$	−5.78409e6	−0.277849
$847$	2.41025e7	1.15439
$848$	1.74670e7	0.834119
$849$	−6.72395e6	−0.320151
$850$	3.41730e6	0.162232
$851$	−4.15635e7	−1.96738
$852$	−1.08697e7	−0.513000
$853$	1.06206e7	0.499779	0.249889	−	0.968274i	$-0.419606\pi$
$853$	1.06206e7	0.499779	0.249889	+	0.968274i	$0.419606\pi$
$854$	−1.91150e7	−0.896872
$855$	−4.66197e6	−0.218099
$856$	−5.63728e6	−0.262957
$857$	−2.58569e7	−1.20261	−0.601304	−	0.799020i	$-0.705352\pi$
$857$	−2.58569e7	−1.20261	−0.601304	+	0.799020i	$0.705352\pi$
$858$	7.68253e6	0.356275
$859$	−9.62997e6	−0.445289	−0.222644	−	0.974900i	$-0.571469\pi$
$859$	−9.62997e6	−0.445289	−0.222644	+	0.974900i	$0.571469\pi$
$860$	5.05374e6	0.233006
$861$	6.07150e6	0.279118
$862$	−2.16006e7	−0.990140
$863$	7.06988e6	0.323136	0.161568	−	0.986862i	$-0.448345\pi$
$863$	7.06988e6	0.323136	0.161568	+	0.986862i	$0.448345\pi$
$864$	4.07081e6	0.185522
$865$	4.36808e6	0.198496
$866$	−3.21602e7	−1.45722
$867$	7.28397e6	0.329095
$868$	−5.19117e6	−0.233865
$869$	−7.40444e7	−3.32616
$870$	1.26556e7	0.566872
$871$	4.93701e6	0.220505
$872$	2.15575e6	0.0960078
$873$	3.08567e6	0.137029
$874$	−5.37827e7	−2.38157
$875$	1.04622e6	0.0461960
$876$	−5.69719e6	−0.250842
$877$	1.37524e7	0.603781	0.301890	−	0.953343i	$-0.402382\pi$
$877$	1.37524e7	0.603781	0.301890	+	0.953343i	$0.402382\pi$
$878$	−1.99041e7	−0.871378
$879$	−5.28345e6	−0.230646
$880$	2.30811e7	1.00473
$881$	−2.92730e7	−1.27066	−0.635328	−	0.772242i	$-0.719135\pi$
$881$	−2.92730e7	−1.27066	−0.635328	+	0.772242i	$0.719135\pi$
$882$	−6.98509e6	−0.302344
$883$	1.86050e7	0.803022	0.401511	−	0.915854i	$-0.368485\pi$
$883$	1.86050e7	0.803022	0.401511	+	0.915854i	$0.368485\pi$
$884$	2.24045e6	0.0964285
$885$	−1.61194e6	−0.0691814
$886$	2.93950e7	1.25803
$887$	3.85656e7	1.64585	0.822926	−	0.568148i	$-0.192340\pi$
$887$	3.85656e7	1.64585	0.822926	+	0.568148i	$0.192340\pi$
$888$	−1.17872e7	−0.501625
$889$	−1.97351e7	−0.837502
$890$	3.59342e6	0.152066
$891$	−4.73581e6	−0.199848
$892$	−502788.	−0.0211579
$893$	2.34933e7	0.985859
$894$	3.26909e7	1.36799
$895$	−1.53652e7	−0.641181
$896$	1.28339e7	0.534058
$897$	−5.07782e6	−0.210715
$898$	−1.95784e7	−0.810190
$899$	3.67294e7	1.51571
$900$	858938.	0.0353473
$901$	−1.06703e7	−0.437890
$902$	5.08890e7	2.08261
$903$	−7.17998e6	−0.293024
$904$	1.10679e7	0.450446
$905$	1.42495e7	0.578334
$906$	−7.00175e6	−0.283391
$907$	−2.62084e7	−1.05785	−0.528923	−	0.848670i	$-0.677404\pi$
$907$	−2.62084e7	−1.05785	−0.528923	+	0.848670i	$0.677404\pi$
$908$	2.42870e6	0.0977596
$909$	−4.09922e6	−0.164548
$910$	1.97962e6	0.0792462
$911$	−4.34825e7	−1.73587	−0.867937	−	0.496674i	$-0.834554\pi$
$911$	−4.34825e7	−1.73587	−0.867937	+	0.496674i	$0.834554\pi$
$912$	−2.65020e7	−1.05510
$913$	−6.57305e7	−2.60970
$914$	1.82878e7	0.724098
$915$	9.17915e6	0.362451
$916$	−2.46097e7	−0.969097
$917$	−4.27488e6	−0.167881
$918$	−3.98593e6	−0.156107
$919$	−9.36547e6	−0.365798	−0.182899	−	0.983132i	$-0.558548\pi$
$919$	−9.36547e6	−0.365798	−0.182899	+	0.983132i	$0.558548\pi$
$920$	−8.77998e6	−0.341998
$921$	7.52226e6	0.292213
$922$	3.58425e7	1.38858
$923$	1.20299e7	0.464792
$924$	7.38021e6	0.284373
$925$	−7.78115e6	−0.299013
$926$	3.81350e7	1.46149
$927$	−5.07476e6	−0.193962
$928$	4.48852e7	1.71093
$929$	7.22351e6	0.274605	0.137303	−	0.990529i	$-0.456157\pi$
$929$	7.22351e6	0.274605	0.137303	+	0.990529i	$0.456157\pi$
$930$	7.19443e6	0.272765
$931$	2.83714e7	1.07277
$932$	303935.	0.0114615
$933$	9.56677e6	0.359800
$934$	1.00597e7	0.377328
$935$	−1.40999e7	−0.527458
$936$	−1.44005e6	−0.0537263
$937$	2.98628e7	1.11117	0.555587	−	0.831458i	$-0.312494\pi$
$937$	2.98628e7	1.11117	0.555587	+	0.831458i	$0.312494\pi$
$938$	1.36878e7	0.507955
$939$	1.38758e7	0.513565
$940$	−4.32849e6	−0.159778
$941$	8.04485e6	0.296172	0.148086	−	0.988974i	$-0.452689\pi$
$941$	8.04485e6	0.296172	0.148086	+	0.988974i	$0.452689\pi$
$942$	1.63437e7	0.600099
$943$	−3.36354e7	−1.23174
$944$	−9.16343e6	−0.334679
$945$	−1.22032e6	−0.0444521
$946$	−6.01798e7	−2.18637
$947$	−1.71255e7	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$947$	−1.71255e7	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$948$	−1.56642e7	−0.566091
$949$	6.30534e6	0.227270
$950$	−1.00687e7	−0.361964
$951$	3.82012e6	0.136970
$952$	−5.50379e6	−0.196820
$953$	−1.12629e7	−0.401715	−0.200857	−	0.979620i	$-0.564373\pi$
$953$	−1.12629e7	−0.401715	−0.200857	+	0.979620i	$0.564373\pi$
$954$	−7.74034e6	−0.275353
$955$	6.35344e6	0.225424
$956$	6.75884e6	0.239181
$957$	−5.22177e7	−1.84305
$958$	−3.90939e7	−1.37624
$959$	1.14091e7	0.400595
$960$	−417310.	−0.0146144
$961$	−7.74932e6	−0.270679
$962$	−1.47231e7	−0.512936
$963$	4.34059e6	0.150829
$964$	−1.95478e7	−0.677493
$965$	690383.	0.0238656
$966$	−1.40782e7	−0.485404
$967$	2.32059e7	0.798054	0.399027	−	0.916939i	$-0.369348\pi$
$967$	2.32059e7	0.798054	0.399027	+	0.916939i	$0.369348\pi$
$968$	−3.78671e7	−1.29889
$969$	1.61897e7	0.553897
$970$	6.66429e6	0.227418
$971$	6.37633e6	0.217031	0.108516	−	0.994095i	$-0.465390\pi$
$971$	6.37633e6	0.217031	0.108516	+	0.994095i	$0.465390\pi$
$972$	−1.00187e6	−0.0340129
$973$	−2.49299e7	−0.844186
$974$	−3.25554e7	−1.09958
$975$	−950625.	−0.0320256
$976$	5.21811e7	1.75343
$977$	−3.20525e7	−1.07430	−0.537150	−	0.843487i	$-0.680499\pi$
$977$	−3.20525e7	−1.07430	−0.537150	+	0.843487i	$0.680499\pi$
$978$	−1.41152e7	−0.471888
$979$	−1.48266e7	−0.494408
$980$	−5.22726e6	−0.173864
$981$	−1.65988e6	−0.0550687
$982$	−3.38374e6	−0.111974
$983$	−1.36173e7	−0.449477	−0.224738	−	0.974419i	$-0.572153\pi$
$983$	−1.36173e7	−0.449477	−0.224738	+	0.974419i	$0.572153\pi$
$984$	−9.53886e6	−0.314057
$985$	2.20947e7	0.725601
$986$	−4.39494e7	−1.43966
$987$	6.14960e6	0.200934
$988$	−6.60127e6	−0.215147
$989$	3.97763e7	1.29310
$990$	−1.02282e7	−0.331674
$991$	−1.89939e7	−0.614369	−0.307184	−	0.951650i	$-0.599387\pi$
$991$	−1.89939e7	−0.614369	−0.307184	+	0.951650i	$0.599387\pi$
$992$	2.55162e7	0.823260
$993$	−1.31738e7	−0.423973
$994$	3.33528e7	1.07069
$995$	9.36596e6	0.299912
$996$	−1.39053e7	−0.444154
$997$	−4.99689e7	−1.59207	−0.796034	−	0.605252i	$-0.793073\pi$
$997$	−4.99689e7	−1.59207	−0.796034	+	0.605252i	$0.793073\pi$
$998$	−5.25590e7	−1.67040
$999$	9.07593e6	0.287725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	195.6.a.d.1.4	✓	4
3.2	odd	2		585.6.a.e.1.1		4
5.4	even	2		975.6.a.g.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.6.a.d.1.4	✓	4	1.1	even	1	trivial
585.6.a.e.1.1		4	3.2	odd	2
975.6.a.g.1.1		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 195 = 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	195.a (trivial)

\(f(q)\)	\(=\)	\(q+6.99762 q^{2} -9.00000 q^{3} +16.9667 q^{4} +25.0000 q^{5} -62.9786 q^{6} +66.9584 q^{7} -105.197 q^{8} +81.0000 q^{9} +174.940 q^{10} -721.813 q^{11} -152.700 q^{12} +169.000 q^{13} +468.549 q^{14} -225.000 q^{15} -1279.07 q^{16} +781.362 q^{17} +566.807 q^{18} -2302.21 q^{19} +424.167 q^{20} -602.625 q^{21} -5050.97 q^{22} +3338.48 q^{23} +946.777 q^{24} +625.000 q^{25} +1182.60 q^{26} -729.000 q^{27} +1136.06 q^{28} -8038.05 q^{29} -1574.46 q^{30} -4569.45 q^{31} -5584.10 q^{32} +6496.31 q^{33} +5467.67 q^{34} +1673.96 q^{35} +1374.30 q^{36} -12449.8 q^{37} -16110.0 q^{38} -1521.00 q^{39} -2629.94 q^{40} -10075.1 q^{41} -4216.94 q^{42} +11914.5 q^{43} -12246.8 q^{44} +2025.00 q^{45} +23361.4 q^{46} -10204.7 q^{47} +11511.6 q^{48} -12323.6 q^{49} +4373.51 q^{50} -7032.26 q^{51} +2867.37 q^{52} -13656.0 q^{53} -5101.26 q^{54} -18045.3 q^{55} -7043.85 q^{56} +20719.8 q^{57} -56247.2 q^{58} +7164.16 q^{59} -3817.50 q^{60} -40796.2 q^{61} -31975.2 q^{62} +5423.63 q^{63} +1854.71 q^{64} +4225.00 q^{65} +45458.7 q^{66} +29213.1 q^{67} +13257.1 q^{68} -30046.3 q^{69} +11713.7 q^{70} +71183.1 q^{71} -8520.99 q^{72} +37309.7 q^{73} -87119.2 q^{74} -5625.00 q^{75} -39060.8 q^{76} -48331.4 q^{77} -10643.4 q^{78} +102581. q^{79} -31976.6 q^{80} +6561.00 q^{81} -70501.6 q^{82} +91063.1 q^{83} -10224.6 q^{84} +19534.0 q^{85} +83373.1 q^{86} +72342.4 q^{87} +75932.8 q^{88} +20540.8 q^{89} +14170.2 q^{90} +11316.0 q^{91} +56642.9 q^{92} +41125.0 q^{93} -71408.5 q^{94} -57555.1 q^{95} +50256.9 q^{96} +38094.6 q^{97} -86235.7 q^{98} -58466.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} - 36 q^{3} + 9 q^{4} + 100 q^{5} + 27 q^{6} - 87 q^{7} - 183 q^{8} + 324 q^{9} - 75 q^{10} - 631 q^{11} - 81 q^{12} + 676 q^{13} + 1344 q^{14} - 900 q^{15} - 351 q^{16} - 599 q^{17} - 243 q^{18}+ \cdots - 51111 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 - 36 * q^3 + 9 * q^4 + 100 * q^5 + 27 * q^6 - 87 * q^7 - 183 * q^8 + 324 * q^9 - 75 * q^10 - 631 * q^11 - 81 * q^12 + 676 * q^13 + 1344 * q^14 - 900 * q^15 - 351 * q^16 - 599 * q^17 - 243 * q^18 - 4 * q^19 + 225 * q^20 + 783 * q^21 - 1920 * q^22 + 4459 * q^23 + 1647 * q^24 + 2500 * q^25 - 507 * q^26 - 2916 * q^27 - 1048 * q^28 - 5274 * q^29 + 675 * q^30 + 2642 * q^31 - 199 * q^32 + 5679 * q^33 + 15794 * q^34 - 2175 * q^35 + 729 * q^36 - 30189 * q^37 - 26408 * q^38 - 6084 * q^39 - 4575 * q^40 - 8947 * q^41 - 12096 * q^42 - 15928 * q^43 - 29676 * q^44 + 8100 * q^45 - 14272 * q^46 - 28242 * q^47 + 3159 * q^48 - 52335 * q^49 - 1875 * q^50 + 5391 * q^51 + 1521 * q^52 - 38021 * q^53 + 2187 * q^54 - 15775 * q^55 - 2480 * q^56 + 36 * q^57 - 115014 * q^58 - 17200 * q^59 - 2025 * q^60 - 67669 * q^61 - 99568 * q^62 - 7047 * q^63 - 69455 * q^64 + 16900 * q^65 + 17280 * q^66 + 8832 * q^67 - 45054 * q^68 - 40131 * q^69 + 33600 * q^70 + 128633 * q^71 - 14823 * q^72 - 48706 * q^73 + 35026 * q^74 - 22500 * q^75 - 66036 * q^76 - 25533 * q^77 + 4563 * q^78 + 72799 * q^79 - 8775 * q^80 + 26244 * q^81 - 191810 * q^82 + 15868 * q^83 + 9432 * q^84 - 14975 * q^85 + 247924 * q^86 + 47466 * q^87 + 5204 * q^88 - 45713 * q^89 - 6075 * q^90 - 14703 * q^91 + 271408 * q^92 - 23778 * q^93 - 51544 * q^94 - 100 * q^95 + 1791 * q^96 - 148949 * q^97 + 10289 * q^98 - 51111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more